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## Topic D: Partitioning and extending segments and parameterization of lines

# Midpoint formula

CCSS.Math:

## Video transcript

Let's say I have the point
3 comma negative 4. So that would be 1, 2,
3, and then down 4. 1, 2, 3, 4. So that's 3 comma negative 4. And I also had the
point 6 comma 1. So 1, 2, 3, 4, 5, 6 comma 1. So just like that. 6 comma 1. In the last video, we figured
out that we could just use the Pythagorean theorem if we
wanted to figure out the distance between these
two points. We just drew a triangle there
and realized that this was the hypotenuse. In this video, we're going to
try to figure out what is the coordinate of the point that
is exactly halfway between this point and that point? So this right here is kind of
the distance, the line that connects them. Now what is the coordinate of
the point that is exactly halfway in between the two? What is this coordinate
right here? It's something comma
something. And to do that-- let me draw
it really big here. Because I think you're going to
find out that it's actually pretty straightforward. At first it seems like a
really tough problem. Gee, let me use the distance
formula with some variables. But you're going to see, it's
actually one of the simplest things you'll learn in
algebra and geometry. So let's say that this is
my triangle right there. This right here is the
point 6 comma 1. This down here is the point
3 comma negative 4. And we're looking for the point
that is smack dab in between those two points. What are its coordinates? It seems very hard at first. But
it's easy when you think about it in terms of just the
x and the y coordinates. What's this guy's x-coordinate
going to be? This line out here represents
x is equal to 6. This over here-- let me do it
in a little darker color-- this over here represents
x is equal to 6. This over here represents
x is equal to 3. What will this guy's
x-coordinate be? Well, his x-coordinate is
going to be smack dab in between the two x-coordinates. This is x is equal to 3, this
is x is equal to 6. He's going to be right
in between. This distance is going to be
equal to that distance. His x-coordinate is going
to be right in between the 3 and the 6. So what do we call the number
that's right in between the 3 and the 6? Well we could call that the
midpoint, or we could call it the mean, or the average,
or however you want to talk about it. We just want to know, what's
the average of 3 and 6? So to figure out this point,
the point halfway between 3 and 6, you literally just figure
out, 3 plus 6 over 2. Which is equal to 4.5. So this x-coordinate
is going to be 4.5. Let me draw that
on this graph. 1, 2, 3, 4.5. And you see, it's smack
dab in between. That's its x-coordinate. Now, by the exact same logic,
this guy's y-coordinate is going to be smack dab between y
is equal to negative 4 and y is equal to 1. So it's going to be right
in between those. So this is the x right there. The y-coordinate is going to be
right in between y is equal to negative 4 and
y is equal to 1. So you just take the average. 1 plus negative 4 over 2. That's equal to negative 3
over 2 or you could say negative 1.5. So you go down 1.5. It is literally right there. So just like that. You literally take the average
of the x's, take the average of the y's, or maybe I should
say the mean to be a little bit more specific. A mean of only two points. And you will get the midpoint
of those two points. The point that's equidistant
from both of them. It's the midpoint of the line
that connects them. So the coordinates are 4.5
comma negative 1.5. Let's do a couple
more of these. These, actually, you're going
to find are very, very straightforward. But just to visualize
it, let me graph it. Let's say I have the point
4, negative 5. So 1, 2, 3, 4. And then go down 5. 1, 2, 3, 4, 5. So that's 4, negative 5. And I have the point
8 comma 2. So 1, 2, 3, 4, 5,
6, 7, 8 comma 2. 8 comma 2. So what is the coordinate
of the midpoint of these two points? The point that is smack
dab in between them? Well, we just average the
x's, average the y's. So the midpoint is going to be--
the x values are 8 and 4. It's going to be 8
plus 4 over 2. And the y value is going to be--
well, we have a 2 and a negative 5. So you get 2 plus negative
5 over 2. And what is this equal to? This is 12 over 2, which is 6
comma 2 minus 5 is negative 3. Negative 3 over 2
is negative 1.5. So that right there
is the midpoint. You literally just average the
x's and average the y's, or find their means. So let's graph it,
just to make sure it looks like midpoint. 6, negative 5. 1, 2, 3, 4, 5, 6. Negative 1.5. Negative 1, negative 1.5. Yep, looks pretty good. It looks like it's equidistant
from this point and that point up there. Now that's all you
have to remember. Average the x, or take the mean
of the x, or find the x that's right in between
the two. Average the y's. You've got the midpoint. What I'm going to show you now
is what's in many textbooks. They'll write, oh, if I have
the point x1 y1, and then I have the point-- actually, I'll
just stick it in yellow. It's kind of painful to switch
colors all the time-- and then I have the point x2 y2, many
books will give you something called the midpoint formula. Which once again, I think is
kind of silly to memorize. Just remember, you
just average. Find the x in between, find
the y in between. So midpoint formula. What they'll really say is the
midpoint-- so maybe we'll say the midpoint x-- or maybe
I'll call it this way. I'm just making up notation. The x midpoint and the y
midpoint is going to be equal to-- and they'll give you this
formula. x1 plus x2 over 2, and then y1 plus y2 over 2. And it looks like something
you have to memorize. But all you have to
say is, look. That's just the average,
or the mean, of these two numbers. I'm adding the two together,
dividing by two, adding these two together, dividing by two. And then I get the midpoint. That's all the midpoint
formula is.